BT003W · Bertrand theorem

mod_eq_bounded_unique

Stable checked-use theorem · independently kernel verified

Two balanced-congruent values below the same modulus are equal.

Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved. Exact original first-admission records.

Statement with defined notation

∀ m. ∀ a. ∀ b. Lt(a,m) → Lt(b,m) → ModEq(m,a,b) → a = b

Every purple notation token opens its conservative definition. Expanding the displayed statement recovers the exact first-order Peano-arithmetic formula checked by the unchanged kernel.

Definitions used by this theorem

In the theorem statement

PD0002 Lt PD0008 ModEq

3 occurrences

In local proof propositions

none

0 occurrences

Exact expanded native-PA statement
forall m a b. (exists ha. ha + S a = m) -> (exists hb. hb + S b = m) -> (exists u v. a + m * u = b + m * v) -> a = b

Proof neighborhood

Direct theorem prerequisites

BT0002 add_comm BT001U division_remainder_unique

Direct theorem dependents

BT003X mod_eq_to_remainder_decomposition

Definition-aware tactic body

Only local propositions introduced by have or suffices are compacted. Every changed line has an exact-AST conservative-expansion receipt; the kernel still receives the immutable original tactic script.

Read the argument

Proof checkpoints

28 script commands · 7 reading checkpoints · 3 local claims

This is a reading aid, not a new proof or a proof-tree certificate. Checkpoint groups are consecutive commands, not inferred branch boundaries. Every step links to the preserved script.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

Named ingredients (2)
add commdivision remainder unique
01Fix variables and assumptionsL1–6

Work with arbitrary variables or the premises of the current implication.

  1. L1
    intro m
  2. L2
    intro a
  3. L3
    intro b
  4. L4
    intro ha
  5. L5
    intro hb
  6. L6
    intro hab
02Separate the logical casesL7–8

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L7
    cases hab
  2. L8
    cases hab_witness
03Establish hdaL9–10

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply add comm.

  1. L9
    have hda : a + m * x = m * x + a
  2. L10
    apply add_comm
04Establish hdbL11–20

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply add comm.

  1. L11
    have hdb : a + m * x = m * x1 + b
  2. L12
    trans b + m * x1
  3. L13
    exact hab_witness_witness
  4. L14
    apply add_comm
  5. L15
    specialize division_remainder_unique m
  6. L16
    specialize division_remainder_unique (a + m * x)
  7. L17
    specialize division_remainder_unique x
  8. L18
    specialize division_remainder_unique a
  9. L19
    specialize division_remainder_unique x1
  10. L20
    specialize division_remainder_unique b
05Establish huniqL21–26

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply division remainder unique.

  1. L21
    have huniq : x = x1 /\ a = b
  2. L22
    apply division_remainder_unique
  3. L23
    exact hda
  4. L24
    exact ha
  5. L25
    exact hdb
  6. L26
    exact hb
06Separate the logical casesL27–27

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L27
    cases huniq
07Use earlier factsL28–28

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L28
    exact huniq_right

Library-wide reading audit

Original defined command ledger · 28 lines
  1. 0001intro m
  2. 0002intro a
  3. 0003intro b
  4. 0004intro ha
  5. 0005intro hb
  6. 0006intro hab
  7. 0007cases hab
  8. 0008cases hab_witness
  9. 0009have hda : a + m * x = m * x + a
  10. 0010apply add_comm
  11. 0011have hdb : a + m * x = m * x1 + b
  12. 0012trans b + m * x1
  13. 0013exact hab_witness_witness
  14. 0014apply add_comm
  15. 0015specialize division_remainder_unique m
  16. 0016specialize division_remainder_unique (a + m * x)
  17. 0017specialize division_remainder_unique x
  18. 0018specialize division_remainder_unique a
  19. 0019specialize division_remainder_unique x1
  20. 0020specialize division_remainder_unique b
  21. 0021have huniq : x = x1 /\ a = b
  22. 0022apply division_remainder_unique
  23. 0023exact hda
  24. 0024exact ha
  25. 0025exact hdb
  26. 0026exact hb
  27. 0027cases huniq
  28. 0028exact huniq_right